Structure theory of commutative quandles and medial Latin quandles

In this blogpost, we solve two open problems posed by Bardakov and Elhamdadi [BE26] in the theory of quandle rings. In particular, non-medial commutative quandles obstruct a conjectural structure theorem of [op. cit.]. However, assuming mediality makes the conjecture hold; we deduce this from an equivalence of categories between commutative (resp. Latin) medial quandles and affine modules over the ring of dyadic rationals (resp. integral Laurent polynomials in \(s\) and \(1-s\)).

Update (1/27/26): I have added Corollary 5.5 and Remarks 5.6 and 5.7.

1. Introduction

In 1982, Joyce [Jo82] and Matveev [Ma82] independently introduced nonassociative algebraic structures called quandles to develop complete invariants of knots. A special class of quandles, called medial quandles, is particularly important in this topological setting. Other special classes of quandles, called Latin quandles and commutative quandles, are important to the theory of certain nonassociative rings called quandle rings.

The purpose of this blogpost is to solve two open problems of Bardakov and Elhamdadi [BE26] in the theory of quandle rings. Along the way, we show that the category of commutative (resp. Latin) medial quandles is equivalent to the category of affine modules over the ring of dyadic rationals \(k:=\mathbb{Z}[1/2]\) (resp. the ring \(\Lambda:=\mathbb{Z}[s^{\pm 1},(1-s)^{-1}]\)), and we deduce structure theorems for finitely generated commutative medial quandles and certain cancellative midpoint algebras.

An auxiliary goal of this blogpost is to clarify the relationships quandles have with a family of algebraic structures called cancellative midpoint algebras. Sigmon [Si70] introduced cancellative midpoint algebras under the name “medial means” in 1970. Cancellative midpoint algebras allow for categorifications of certain aspects of convex analysis [ES01, Fr08], and they have connections to (affine) modules over \(k\) [Ba24, Fr08].

1.1. Definitions

First, we recall definitions coming from nonassociative algebra. Recall that a magma is a set \(X\) equipped with a binary operation \(\ast\colon X\to X\) called multiplication. The cardinality of \(X\) is called its order. Given magmas \( (X,\ast_X)\) and \( (Y,\ast_Y)\) and a function \(f\colon X\to Y\), we say that \(f\) is a magma homomorphism if \(f(w\ast_X x)=f(w)\ast_Y f(x)\) for all \(w,x\in X\).

Let \((X,\ast)\) be a magma. For all \(x\in X\), define the right multiplication map \(R_x\colon X\to X\) by \(y\mapsto y\ast x\), and define the left multiplication map \(L_x\colon X\to X\) by \(y\mapsto x\ast y\).

• \(X\) is called Latin or a left quasigroup if for all \(x\in X\), the left multiplication map \(L_x\) is invertible. If in addition the right multiplication maps \(R_x\) are all invertible, then we call \(X\) a quasigroup.
• \(X\) is called a rack if for all \(x\in X\), the right multiplication map \(R_x\) is a magma automorphism. In particular, the multiplication \(\ast\) is right-distributive: \[(x\ast y)\ast z=(x\ast z)\ast (y\ast z).\]
• \(X\) is called idempotent if \(x\ast x=x\) for all \(x\in X\). Quandles are idempotent racks, and kei are quandles such that all right multiplication maps \(R_x\) are involutions.
• Note that the Cartesian product \(X\times X\) is a magma. We call \(X\) medial if the multiplication \(\ast\colon X\times X\to X\) is a magma homomorphism; that is,\[(w\ast x)\ast(y\ast z)=(w\ast y)\ast(x\ast z)\] for all \(w,x,y,z\in X\).
• \(X\) is called commutative if \(x\ast y=y\ast x\) for all \(x,y\in X\). That is, \(L_x=R_x\) for all \(x\in X\).
• Midpoint algebras are idempotent medial commutative magmas.
• \(X\) is called cancellative if for all \(x\in X\), the left multiplication map \(L_x\) is injective.
• \(X\) is called distributive if for all \(x\in X\), the left and right multiplication maps \(L_x,R_x\) are magma endomorphisms.

The prototypical example of a quandle is the conjugation quandle of a group \(G\), defined to be the pair \(\mathrm{Conj}(G):=(G,\ast)\) with \[g\ast h:=hgh^{-1}.\] A common class of kei are core quandles of groups \(G\), which are defined to be \(\mathrm{Core}(G):=(G,\ast)\) with \[g\ast h:=hg^{-1}h.\] On the other hand, Alexander quandles are important examples of medial quandles. Given an automorphism \(\varphi\) of an abelian group \(A\), the Alexander quandle \(\mathrm{Alex}(A,\varphi)=(A,\ast)\) is defined by \[x\ast y:=\varphi(x)+(\mathrm{id}-\varphi)(y).\]

1.2. Organization of this blogpost

In Section 2, we record some lemmas and discuss a family of commutative medial quandles that we call averaging quandles. Averaging quandles turn out to model all commutative medial quandles, as we show later on in Section 4.

In Section 3, we solve Question 7.1 of [BE26]. In particular, we discuss an obstruction to a general structure theorem for commutative quandles, and we completely characterize both commutative and noncommutative quandles whose dual quandles are commutative.

In Section 4, we provide new proofs characterizing all medial Latin quandles as certain Alexander quandles and, in particular, all commutative medial quandles as averaging quandles.

In Section 5, we strengthen the results of Section 4 by showing that “taking the Alexander quandle” (resp. the averaging quandle) defines an equivalence of categories from affine modules over \(\Lambda=\mathbb{Z}[s^{\pm 1}, (1-s)^{-1}]\) (resp. \(k=\mathbb{Z}[1/2]\)) to Latin (resp. commutative) medial quandles.

In Section 6, we completely describe free objects in the categories studied in Section 5. As an application, we deduce structure theorems for finitely generated commutative medial quandles and certain cancellative midpoint algebras. This solves Question 7.3 and resolves a conjectural structure theorem in Question 7.1 of [BE26].

2. Preliminaries

In this section, we record some auxiliary results and consider a class of commutative medial quandles called averaging quandles \(M_{\mathrm{avg}}\). (Later in this blogpost, we will show that all commutative medial quandles are isomorphic to some averaging quandle.)

2.1. Preliminary results

Commutativity turns out to be a very strong condition for quandles. The following lemmas are straightforward and left to the reader.

Lemma 2.1.

• Every Latin rack is an idempotent quasigroup and, in particular, a quandle.
• Every commutative rack is distributive and Latin—in particular, a quasigroup and a quandle.
• Every distributive quasigroup is a Latin quandle.

Lemma 2.2 (cf. [Ba24]).

• Let \( (X,\ast)\) be a finite commutative magma. Then \(X\) is cancellative if and only if it is Latin.
• Commutative medial quandles are the same as Latin midpoint algebras. In particular, finite commutative medial quandles are the same as finite cancellative midpoint algebras.

Remark 2.3. Infinite cancellative midpoint algebras are not necessarily quasigroups; in particular, Lemma 2.1 implies they are not necessarily quandles. See Remark 2.7 for a counterexample.

2.2. Averaging quandles

Henceforth, \(k:=\mathbb{Z}[\frac{1}{2}]\) denotes the ring of dyadic rationals. All abelian groups are denoted additively.

In this section, we define a class of commutative medial quandles called averaging quandles. The quandles in parts (1) and (2) of [BE26, Ex. 5.1] are special classes of averaging quandles.

Definition 2.5. Let \(A\) be a unital ring in which \(2\) is invertible, and let \(M\) be a left \(A\)-module. Define a quandle operation on \(M\) by averaging: \[x\ast y:=\frac{1}{2}(x+y).\] Then \( M_{\mathrm{avg}}:=(M,\ast) \) is a commutative quandle called the averaging quandle on \(M\).

In particular, let \(M=\mathbb{Z}/(2n+1)\) be the cyclic group of order \(2n+1\) with \(n\geq 0\). Then we denote the corresponding averaging quandle by \(C_{2n+1}:=M_{\mathrm{avg}}\). (Since \(2^{-1}=n+1\) in \(M\), this definition of \(C_{2n+1}\) coincides with the one from [BE26].)

Remark 2.6. Averaging quandles are a special class of so-called weighted average quandles (see [BT26]), which are themselves a special class of Alexander quandles. Indeed, if \( M_{\mathrm{avg}}\) is an averaging quandle, then multiplication by \(1/2\) defines an automorphism of \(M\) whose corresponding Alexander quandle is precisely \(M_{\mathrm{avg}}\). In particular, all averaging quandles are medial. In general, though, Alexander quandles are not necessarily commutative.

Remark 2.7. Contrary to Example 5.1(3) and Question 7.3 of [BE26], the subset \(X:=\{m/2^n \mid m,n\in\mathbb{Z}_{\geq 0}\}\subset k\) is not a subquandle of \(k_{\mathrm{avg}}\). Indeed, the right multiplication map \( R_1\) does not restrict to a permutation of \(X\).

3. First solutions

In this section, we solve Question 7.1 of Bardakov and Elhamdadi [BE26]. Of the question’s three parts, the second can be solved fairly quickly.

Proposition 3.1 (cf. [BE26, Question 7.1]). Let \(G\) be a group. Then:
(1) \(\mathrm{Conj}(G)\) is commutative if and only if \(G\) is the trivial group.
(2) \(\mathrm{Core}(G)\) is commutative if and only if the exponent of \(G\) is \(\mathrm{exp}(G)=3\).

Proof. (1): If \(\mathrm{Conj}(G)\) is commutative, then \[g=g\ast e=e\ast g=e\] for all \(g\in G\), so \(G=\{e\}\). The converse is trivial.

(2): If \(\mathrm{Core}(G)\) is commutative, then for all \(g\in G\), we have \[g^2=e\ast g=g\ast e=g^{-1}\] and, hence, \(g^3=e\). Conversely, suppose \(\mathrm{exp}(G)=3\). Then for all \(g,h\in G\), we have \( (hg^{-1})^2 = (hg^{-1})^{-1} \), so \[ g\ast h= hg^{-1}h=(hg^{-1})^2 g = (hg^{-1})^{-1} g = gh^{-1}g=h\ast g, \] as desired. QED.

3.1. An obstruction to structure theorems

The first part of [BE26, Question 7.1] asks whether every finite commutative quandle can be written as a direct product of averaging quandles of the form \(C_{2n+1}\). In this section, we show that the question has a negative answer. Later in this blogpost (see Theorem 6.3), we will give an additional assumption (namely, mediality) under which this question has a positive answer.

To give a negative answer to [BE26, Question 7.1], it suffices to construct a finite commutative quandle that is not medial. This is due to Remark 2.6 and the fact that direct products of medial quandles are medial. Indeed, such quandles have already appeared in the literature. In 1981, Kepka and Němec [KN81, Thm. 12.4] (see [St15A, Ex. 3.4]) constructed non-medial distributive quasigroups (hence quandles by Lemma 2.1) of order 81. (They also showed that these are the smallest such examples.) Of their constructions, the following is in fact a commutative quandle.

Given an abelian group \(A\) and a function \(f\colon A^3\to A\), we say that \(f\) is triadditive if for all \(x,y\in A\), the restrictions \(f(-,x,y)\), \(f(x,-,y)\), and \(f(x,y,-)\) are endomorphisms of \(A\). In particular, let \(\mathbb{Z}/3\) denote the cyclic group of order 3, let \(A:= (\mathbb{Z}/3)^4\), and let \(e_1,e_2,e_3,e_4\) be the canonical generators of \(A\). Define a triadditive function \(f\) by extending

Make \(A\) into a commutative Moufang loop via the operation \[x\cdot y:=x+y+f(x,y,x-y).\] (This commutative Moufang loop is called \(L(1)\) in [KN81] and \((G_1,\cdot)\) in [St15A].) Then \(A\) is a quandle with respect to the operation \[x\ast y := (-x)\cdot(-y).\] (This quandle is called \(D(1)\) in [KN81].) Since \((A,\cdot)\) is commutative, it follows that \((A,\ast)\) is also commutative.

Although it is already shown in [KN81, St15A] that \((A,\ast)\) is not medial, we provide a direct verification for the reader’s convenience:

3.2. Cocommutative quandles

In this section, we solve the third part of [BE26, Question 7.1]. In the following, let \((X,\ast)\) be a rack, denoted simply by \(X\). Recall that the dual rack of \(X\) is the rack \(X^{\mathrm{op}}:=(X,\overline{\ast})\), where \[x\overline{\ast}y:=R_y^{-1}(x).\] (For a proof that \(X^{\mathrm{op}}\) is a rack, see [BT26].) The name is justified because \((X^{\mathrm{op}})^{\mathrm{op}}=X\).

We will say that \(X\) is cocommutative if its dual rack \(X^{\mathrm{op}}\) is commutative. Below, we provide a necessary and sufficient condition under which \(X\) is cocommutative. In particular, we completely describe racks that are both commutative and cocommutative; this solves part (3) of [BE26, Question 7.1]. These results appear to be new.

Theorem 3.2. Let \((X,\ast)\) be a rack. The following are equivalent:
(1) \(X\) is cocommutative.
(2) \(X\) is a cocommutative quandle.
(3) For all \(x\in X\), the left multiplication map \(L_x\) is an involution.

Proof. First, note that (3) is equivalent to the statement that \[x\ast(x\ast y)=y\] for all \(x,y\in X\).

”\((1)\implies(2)\)”: If \(X\) is cocommutative, then Lemma 2.1 implies that \(X^{\mathrm{op}}\) is a quandle. That is, \(R_x^{-1}(x)=x\) for all \(x\in X\). Applying \(R_x\) to both sides shows that \(X\) is also a quandle.

”\((2)\implies(3)\)”: Assume that \(X\) is a cocommutative quandle. Given \(x,y\in X\), let \(z:=R_y^{-1}(x)\). By assumption, \(z=R_x^{-1}(y)\). Since \(X\) is a quandle, it follows that \[x\ast(x\ast y)=(z\ast y)\ast (x\ast y)=(z\ast x)\ast y=y\ast y=y,\] as desired.

”\((3)\implies(1)\)”: Given \(x,y\in X\), let \(z:=R_y^{-1}(x)\). We have to show that \(z=R_x^{-1}(y)\). Indeed, \[ R_x(z)= R_{R_y(z)}(z) = z\ast(z\ast y)=y, \] where in the last equality we have used the assumption. Since \(R_x\) is invertible, the claim follows. QED.

Corollary 3.3. Every cocommutative rack is a Latin quandle.

Corollary 3.4 (cf. [BE26, Question 7.1]). Let \((X,\ast)\) be a commutative quandle. Then \(X\) is cocommutative if and only if \(X\) is a kei.

Proof. Since \(X\) is commutative, we have \(L_x=R_x\) for all \(x\in X\). Hence, Theorem 3.2 yields the claim. QED.

Remark 3.5. Commutative kei are the same as distributive Steiner quasigroups, which are algebraic structures that correspond to combinatorial designs called Hall triple systems. See [St15T, Sec. 3.4] for further discussion and references on distributive Steiner quasigroups, and see [NP06] for a quandle-theoretic treatment of commutative kei.

4. Classification of medial Latin quandles

In light of Section 3.1, it is natural to ask for necessary and sufficient conditions under which finite quandles decompose as direct products of averaging quandles of the form \(C_{2n+1}\). Later in this blogpost (see Theorem 6.3), we will show that these conditions are precisely commutativity and mediality. To that end, this section completely describes medial Latin quandles and commutative medial quandles. Although the results of this section were already shown in [Ba24, JPSZ15], we provide new, much shorter proofs at the cost of using the Bruck–Murdoch–Toyoda theorem.

Recall from Lemma 2.1 that every Latin quandle (and, hence, every commutative quandle) is a quasigroup. In the medial case, this observation allow us to appeal to the well-known Bruck–Murdoch–Toyoda theorem, which states the following: For every medial quasigroup \( (X,\ast)\), there exists an abelian group \(A\), a fixed element \(c\in A\), and two commuting automorphisms \(\varphi,\psi\) of \(A\) such that \(X\) is isomorphic to the medial quasigroup \( (A,\cdot)\) with multiplication \[a\cdot b := \varphi(a)+\psi(b)+c.\]

Proposition 4.1 ([JPSZ15, Ex. 2.2 and Cor. 3.4]). Every medial Latin quandle is isomorphic to an Alexander quandle \(\mathrm{Alex}(A,\varphi)\) such that the map \(\mathrm{id}-\varphi\) is an automorphism of \(A\).

Proof. By the above discussion, every medial Latin quandle is isomorphic to a quasigroup of the form \( (A,\cdot)\) described above. By assumption, \( (A,\cdot)\) is idempotent, so taking \(a:=0\) and \(b:=0\) above shows that \(c=0\). Therefore, idempotence forces \(\varphi+\psi=\mathrm{id}\). That is, \[a\cdot b=\varphi(a)+(\mathrm{id}-\varphi)(b),\] so \( (A,\cdot)=\mathrm{Alex}(A,\varphi),\) as desired. Finally, since \( (A,\cdot)\) is Latin, the left multiplication maps \(L_a=\varphi(a)+\mathrm{id}-\varphi\) are invertible. Since addition by \(\varphi(a)\) is invertible, it follows that \(\mathrm{id}-\varphi\) is also invertible. QED.

Corollary 4.2 ([Ba24]). Every commutative medial quandle is isomorphic to an averaging quandle.

Proof. By Proposition 4.1, every commutative medial quandle is isomorphic to an Alexander quandle \( \mathrm{Alex}(A,\varphi) \). Commutativity implies that \[\varphi(a)=a\ast 0=0\ast a=a-\varphi(a)\] for all \(a\in A\); that is, \(2\varphi=\mathrm{id}\). Since \(\varphi\) is invertible, it follows that multiplication by \(2\) is invertible. Hence, \(A\) is a \(k\)-module, and \(\varphi\) is multiplication by \(1/2\). QED.

Remark 4.3. The “converses” of Proposition 4.1 and Corollary 4.2 also hold. Precisely, every Alexander quandle \(\mathrm{Alex}(A,\varphi)\) is medial, and if \(\mathrm{id}-\varphi\) is invertible, then \(\mathrm{Alex}(A,\varphi)\) is also Latin. Moreover, every averaging quandle is commutative and medial.

4.1. Constructions

In light of Lemma 2.2, the construction of Bauer in [Ba24] can be viewed as a way to recover the averaging quandle corresponding to a given commutative medial quandle under Theorem 4.2. Although we will not use this construction, it is worth recording in our notation. Namely, given a nonempty commutative quandle \( (X,\ast)\), fix a basepoint \(0\in X\). Bauer showed that the following operations define a \(k\)-module structure on \(X\) such that \(0\) is the additive identity and \(X_{\mathrm{avg}}\cong(X,\ast)\):

Of course, the \(k\)-modules obtained from this construction may have more desirable or familiar forms. The following gives such an example; in fact, we will later show that this example characterizes free commutative medial quandles.

Recall that \(k=\mathbb{Z}[1/2]\) denotes the ring of dyadic rationals. Henceforth, given a set \(X\) and a ring \(R\), let \(R^{(X)}\) denote the free left \(R\)-module generated by \(X\).

Example 4.4. Given a nonempty set \(X\), consider the affine hull \(H_X\) of \(X\) in \(k^{(X)}\): \[H_X:= \left\{\sum^n_{i=0}\lambda_i x_i: n\geq 1,\, \lambda_i\in k,\, x_i\in X,\, \sum^n_{i=0}\lambda_i=1 \right\}\subset k^{(X)}.\] Then \(H_X\) is a subquandle of the averaging quandle \( (k^{(X)})_{\mathrm{avg}}\). To write \(H_X\) itself as an averaging quandle, fix a basepoint \(x_0\in X\), and consider the map

The reader can check that this map is a quandle isomorphism. In particular, if \(\# X=n\) with \(1\leq n<\infty\), then we obtain a quandle isomorphism \(H_X\cong k^{n-1}_{\mathrm{avg}}\).

5. Equivalences of categories

Proposition 4.1 and Corollary 4.2 allow us to give an alternative, commutative-algebraic perspective on the theory of medial Latin quandles. In particular, since cancellative midpoint algebras are closely related to affine modules over \(k\) (see, for example, [Ba24, Fr08]), Lemma 2.2 suggests that commutative medial quandles should be related to these modules. We will formalize this idea using an equivalence of categories.

Given a ring \(R\), define a category \(\mathsf{AffMod}_R\) as follows. The objects consist of the empty set and all left \(R\)-modules. The morphisms are affine transformations (that is, sums of \(R\)-module homomorphisms and constant functions). We call \(\mathsf{AffMod}_R\) the category of (left) affine modules over \(R\).¹

5.1. Medial Latin quandles

Henceforth, let \(\mathsf{MLQnd}\) be the category of medial Latin quandles and quandle homomorphisms, and denote the ring of integral Laurent polynomials in \(s\) and \(1-s\) by \[\Lambda:=\mathbb{Z}[s^{\pm 1},(1-s)^{-1}].\] Building upon Proposition 4.1, we will show that \(\mathsf{AffMod}_{\Lambda}\) and \(\mathsf{MLQnd}\) are equivalent.

Note that the data of an object \(M\) in \(\mathsf{AffMod}_{\Lambda}\) is equivalent to the data of an abelian group automorphism \(\varphi\in\mathrm{Aut}(M)\) such that the map \(\mathrm{id}-\varphi\) is invertible; cf. [BT26]. (Explicitly, the correspondence is given by \(s^{\pm 1}\cdot m \leftrightarrow \varphi^{\pm 1}(m)\) and \((1-s)^{\pm 1}\cdot m\leftrightarrow (\mathrm{id}-\varphi)^{\pm 1}(m)\) for all \(m\in M\).) So, define a functor \(\mathrm{Alex}\colon\mathsf{AffMod}_{\Lambda}\to\mathsf{MLQnd}\) on objects by sending \(M\) to the induced Alexander quandle \(\mathrm{Alex}(M,\varphi)\). Let \(\mathrm{Alex}\) fix all morphisms (as set-theoretic maps).

Lemma 5.1. The assignment \(\mathrm{Alex}\colon\mathsf{AffMod}_{\Lambda}\to\mathsf{MLQnd}\) is a functor.

Proof. We only have to show that every affine transformation of \(\Lambda\)-modules \(f\colon M\to N\) is a quandle homomorphism \(\mathrm{Alex}(M,\varphi)\to\mathrm{Alex}(N,\psi)\). If \(M\) is empty, then the claim is trivial. Othewise, \(f\) factorizes as \(f=T+c\) for some \(\Lambda\)-linear map \(T\colon M\to N\) and some constant \(c\in N\). In particular, \(T\circ\varphi = \psi\circ T\), so

for all \(x,y\in M\). Hence, \(f\) is a quandle homomorphism. QED.

Lemma 5.2. Let \(M\) and \(N\) be \(\Lambda\)-modules, and let \(T\colon M\to N\) be a \(\mathbb{Z}[s]\)-linear map. Then \(T\) is \(\Lambda\)-linear.

Proof. Left to the reader; use the facts that \(T\) commutes with \(s\) and \(1-s\) and that multiplication by \(s\) and \(1-s\) are invertible. QED.

Next, we prove the main result of this section.

Theorem 5.3. The functor \(\mathrm{Alex}\colon\mathsf{AffMod}_{\Lambda}\to\mathsf{MLQnd}\) is an equivalence of categories.

Proof. By Lemma 5.1, Proposition 4.1, and the above discussion, \(\mathrm{Alex}\) is an essentially surjective functor. Therefore, it suffices to show that \(\mathrm{Alex}\) is fully faithful. Faithfulness is clear. To show fullness, let \(f\colon \mathrm{Alex}(M,\varphi)\to\mathrm{Alex}(N,\psi)\) be a homomorphism of Alexander quandles satisfying the conditions of Proposition 5.1, and view \(M\) and \(N\) as \(\Lambda\)-modules as in the above discussion. Let \(c:= f(0)\), and define \(T\colon M\to N\) by \(T:= f-c\). Then \(f=T-c\), so we only have to show that \(T\) is \(\Lambda\)-linear. By Lemma 6.2, it suffices to show that \(T\) is a homomorphism of abelian groups such that \(T\circ\varphi=\psi\circ T\). Clearly, \(T(0)=0\).

Since \(f\) is a quandle homomorphism, the reader can verify that \(T\) is also a quandle homomorphism. Equivalently, \[T(\varphi(x-y)+y)=\psi(T(x))+(\mathrm{id}-\psi)(T(y))\] for all \(x,y\in M\). In particular, for all \(z\in M\), taking \((x,y):=(\varphi^{-1}(z),0)\) shows that \(T=\psi\circ T\circ\varphi^{-1}\). Therefore, \(T\circ\varphi=\psi\circ T\), as desired.

Now, define a function \(g\colon M^2\to N\) by \[g(x,y):=T(x+y)-T(x)-T(y)\] for all \(x,y\in M\). It remains to show that \(g\equiv 0\). Since \(\varphi\) is a homomorphism of abelian groups and \(T\) is a quandle homomorphism, we first note that

for all \(a,b,c,d\in M\). Since \(T\) is a quandle homomorphism and \(\psi\) is a homomorphism of abelian groups, it follows that

for all \(a,b,c,d\in M\). In particular, given \(x,y\in M\), let \[a:=\varphi^{-1}(x),\quad b:= 0,\quad c:=0,\quad d:=(\mathrm{id}-\varphi)^{-1}(y).\] Then \(x=a\ast b\) and \( y=c\ast d\), so \[g(x,y)=g(a,c)\ast g(b,d)=g(a,0) \ast g(0,d)=0\ast 0=0\] because \(g(-,0)\equiv 0\) and \(g(0,-)\equiv 0\). Hence, \(T\) is \(\Lambda\)-linear. QED.

5.2. Commutative medial quandles

Let \(\mathsf{CMQnd}\) be the category of commutative medial quandles and quandle homomorphisms, and recall that \(k=\mathbb{Z}[1/2]\) denotes the ring of dyadic rationals. Specializing Theorem 5.3 yields an equivalence of categories from \(\mathsf{AffMod}_{k}\) to \(\mathsf{CMQnd}\). We give more details below.

By restriction of scalars, the ring epimorphism \[\Lambda\twoheadrightarrow\Lambda/(2s-1)\xrightarrow{\sim}k\] induces a fully faithful embedding \(\mathsf{AffMod}_{k}\hookrightarrow \mathsf{AffMod}_{\Lambda}\). The image consists precisely of affine \(\Lambda\)-modules \(M\) such that \((2s-1)M=0\). Composing with the equivalence of categories \(\mathrm{Alex}\colon\mathsf{AffMod}_{\Lambda}\xrightarrow{\sim}\mathsf{MLQnd}\) from Theorem 5.3 shows that \(\mathsf{AffMod}_{k}\) is equivalent to the category of averaging quandles \(M_{\mathrm{avg}}\). By Corollary 4.2, the latter category is equivalent to \(\mathsf{CMQnd}\). Summarizing, we have the following.

Corollary 5.4. The functor \(\mathrm{Alex}\) induces an equivalence of categories \(\mathrm{avg}\colon\mathsf{AffMod}_k\xrightarrow{\sim}\mathsf{CMQnd}\) that sends every affine \(k\)-module \(M\) to its averaging quandle \(M_{\mathrm{avg}}\).

The reader can verify for all rings \(R\), two nonempty objects in \(\mathsf{AffMod}_{R}\) are isomorphic if and only if they are isomorphic as left \(R\)-modules. Therefore, combining Lemma 5.2 with Theorem 5.3 and Corollary 5.4 yields the following.

Corollary 5.5. Two Latin Alexander quandles \(\mathrm{Alex}(M,\varphi)\) and \(\mathrm{Alex}(N,\psi)\) are isomorphic if and only if \(M\) and \(N\) are isomorphic as \(\mathbb{Z}[s]\)-modules. In particular, two averaging quandles are isomorphic if and only if they are isomorphic as \(k\)-modules.

Remark 5.6. In 2003, Nelson [Ne03, Thm. 2.1] proved that if \(\mathrm{Alex}(M,\varphi)\) and \(\mathrm{Alex}(N,\psi)\) are finite Alexander quandles, then they are isomorphic if and only if \((1-s)M\) and \((1-s)N\) are isomorphic as \(\mathbb{Z}[s^{\pm 1}]\)-modules. Since multiplication by \(1-s\) is invertible for Latin Alexander quandles, Corollary 5.5 shows that Nelson’s result also holds if the word “finite” is replaced with “Latin.”

Remark 5.7. Corollary 5.5 fails if we drop the assumption that \(\mathrm{Alex}(M,\varphi)\) and \(\mathrm{Alex}(N,\psi)\) are Latin; this is equivalent to dropping the assumption that \(1-s\) is invertible. For example, Nelson [Ne03] shows that \(\mathbb{Z}/(9,s-4)\) and \(\mathbb{Z}/(9,s-7)\) are isomorphic as Alexander quandles but not as \(\mathbb{Z}[s^{\pm 1}]\)-modules. Accordingly, these \(\mathbb{Z}[s^{\pm 1}]\)-modules are not \(\Lambda\)-modules.

6. Free objects

As an application of Theorem 5.3 and Corollary 5.4, we completely describe free medial Latin quandles and free commutative medial quandles. As another application, we obtain a structure theorem for finitely generated commutative medial quandles.

6.1. Free affine modules

We describe free (left) affine modules over a ring \(R\). Since left affine modules over \(R\) form a variety in the sense of universal algebra, it follows that free objects in \(\mathsf{AffMod}_R\) exist and satisfy the usual universal property. Namely, if \(X\) is a set, then the free (left) affine \(R\)-module generated by \(X\) is the (unique up to isomorphism) affine \(R\)-module \(F_R(X)\) along with an injective map \(\iota\colon X\hookrightarrow F(X)\) such that for all affine \(R\)-modules \(M\) and all set-theoretic functions \(f_0\colon X\to M\), there exists a unique affine transformation \(f\colon F_R(X)\to M\) such that \(f\circ\iota=f_0\).

More explicitly, we can construct affine modules as follows. Let \(X\) be a set. If \(X\) is empty, then \(F_R(X)\) is also empty. Otherwise, fix a basepoint \(z\in X\), and let \(F_R(X):=R^{(X\setminus\{z\})}\) be the free (left) \(R\)-module generated by the set \(X\setminus\{z\}\). Define \(\iota\colon X\hookrightarrow F_R(X)\) by \(z\mapsto 0\) and \(x\mapsto x\) for all \(x\in X\setminus\{z\}\). The reader can verify that \(F_R(X)\) satisfies the universal property of free affine modules with \(f:=T+z\), where \(T\colon F_R(X)\to M\) is the unique linear map obtained by extending the assignment \(x\mapsto f_0(x)-z\) for all \(x\in X\).

6.2. Free medial Latin quandles

Theorem 5.3 and Corollary 5.4 provide equivalences of categories \(\mathrm{Alex}\colon\mathsf{AffMod}_{\Lambda}\xrightarrow{\sim}\mathsf{MLQnd}\) and \(\mathrm{avg}\colon\mathsf{AffMod}_k\xrightarrow{\sim}\mathsf{CMQnd}\). Since equivalences of categories preserve universal properties, it follows that free objects in the target categories are precisely the images of the free objects in the source categories.

More explicitly, let \(X\) be a set. We construct the free medial Latin quandle \(\mathrm{FML}(X)\) and the free commutative medial quandle \(\mathrm{FCM}(X)\) as follows. If \(X\) is empty, then these quandles are also empty. Otherwise, fixing a basepoint \(z\in X\), recall that we have free affine modules \(F_\Lambda(X)\) and \(F_k(X)\). Then \(\mathrm{FML}(X)\) is the Alexander quandle \(\mathrm{Alex}(F_\Lambda(X),\varphi)\), where \(\varphi\) denotes multiplication by \(s\). On the other hand, \(\mathrm{FCM}(X)\) is the averaging quandle \(F_k(X)_{\mathrm{avg}}\). In particular, if \(X\) is a set of cardinality \(1\leq n<\infty\), then \(\mathrm{FCM}(X)\cong k^{n-1}_{\mathrm{avg}}\).

Remark 6.1. Given a nonempty set \(X\), Example 4.4 can be restated as an isomorphism between \(\mathrm{FCM}(X)\) and the affine hull \(H_X\) considered as a subquandle of \(k^{(X)}_{\mathrm{avg}}\). This aligns with the intuition that since \(H_X\) is (in the geometric sense) the smallest affine subset of \(k^{(X)}\) containing \(X\), it should correspond to the free affine \(k\)-module \(F_k(X)\) and its corresponding commutative medial quandle \(\mathrm{FCM}(X)\).

Remark 6.2. In the category of commutative quandles with quandle homomorphisms—which, as shown in Section 3.1, contains \(\mathsf{CMQnd}\) as a strictly smaller subcategory—the above discussion also characterizes free objects with three or fewer generators. Namely, given a set \(X\) of cardinality \(1\leq n\leq 3\), let \(\mathrm{FC}(X)\) denote the free commutative quandle generated by \(X\). By Lemma 2.1, \(\mathrm{FC}(X)\) is distributive. Since \(n\leq 3\), it follows from [JK83, Prop. 3.2] that \(\mathrm{FC}(X)\) is medial. Therefore, we obtain isomorphisms \(\mathrm{FC}(X)\cong\mathrm{FCM}(X)\cong k^{n-1}_{\mathrm{avg}}\). In particular, the \(n=2\) case solves an open problem of Bardakov and Elhamdadi [BE26, Question 7.3]; cf. Remark 2.7. (Note that this argument fails if \(\# X\geq 4\). In this case, we suspect that the the non-medial commutative quandle \((A,\ast)\) from Section 3.1 is a quotient of \(\mathrm{FC}(X)\); if this is true, then \(\mathrm{FC}(X)\) must also be non-medial.)

6.3. Finitely generated commutative medial quandles

As one last application of Corollary 5.4, we obtain a structure theorem for finitely generated commutative medial quandles. In Section 6.2, we found that if \(X\) is a set of cardinality \(1\leq n<\infty\), then the free commutative medial quandle \(\mathrm{FCM}(X)\) is isomorphic to the averaging quandle \(k^{n-1}_{\mathrm{avg}}\). On the other hand, \(k\) is a PID, and all finite \(k\)-modules must have odd order; otherwise, multiplication by \(2\) would not be invertible. Hence, combining Lemma 2.1, Corollary 5.4, and the structure theorem for finitely generated modules over a PID yields the following.

Theorem 6.3. (cf. [BE26, Question 7.1]). Let \(M\) be a nonempty finitely generated rack. Then \(M\) is commutative and medial if and only if \[M\cong k^r_{\mathrm{avg}} \times\prod^n_{i=1} C_{2m_i+1}\] for some nonnegative integers \(r,n,m_1,\dots,m_n\geq 0\).

Remark 6.4. By Lemma 2.2, Theorem 6.3 is also a structure theorem for finitely generated Latin midpoint algebras. Moreover, if \(M\) is finite in Theorem 6.3, then \(r=0\), so we conclude that every finite commutative medial quandle—hence, by Lemma 2.2, every finite cancellative midpoint algebra—is the direct product of averaging quandles of the form \(C_{2n+1}\). This gives a complete characterization of quandles for which the conjecture in the first part of [BE26, Question 7.1] holds.

References

[BE26] V. Bardakov and M. Elhamdadi. Idempotents and powers of ideals in quandle rings, 2026. arXiv:2601.07057

[Ba24] K. J. Bauer, quasigroup midpoint algebras = Z[1/2]-modules, Frank Blog, 2024. https://frank-9976.github.io/midpoint.html (accessed: 1-14-2026).

[BT26] W. Burrows and C. Tuffley, The rack congruence condition and half congruences in racks, to appear in J. Knot Theory Ramifications. doi:10.1142/S0218216525500865.

[ES01] M. H. Escardo and A. K. Simpson, A universal characterization of the closed Euclidean interval. Proc. 16th Annual IEEE Symposium on Logic in Comp. Sci. (2001), 115–125. doi:10.1109/LICS.2001.932488.

[Fr08] P. Freyd, Algebraic real analysis, Theory Appl. Categ. 20 (2008), no. 10, 215–306. MR2425550

[JPSZ15] P. Jedlička, A. Pilitowska, D. Stanovský, and A. Zamojska-Dzienio, The structure of medial quandles, J. Algebra 443 (2015), 300–334. MR3400403

[JK83] J. Ježek and T. Kepka, Notes on distributive groupoids, Comment. Math. Univ. Carolin. 24 (1983), no. 2, 237–249. MR0711262

[Jo82] D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37–65. MR638121

[KN81] T. Kepka and P. Němec, Commutative Moufang loops and distributive groupoids of small orders, Czechoslovak Math. J. 31(106) (1981), no. 4, 633–669. MR0631607

[Ma82] S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78–88, 160. MR672410

[Ne03] S. Nelson, Classification of finite Alexander quandles, Topology Proc. 27 (2003), no. 1, 245–258. MR2048935

[NP06] M. Niebrzydowski and J. H. Przytycki, Burnside kei, Fund. Math. 190 (2006), 211–229. 2232860

[Si70] K. Sigmon, Cancellative medial means are arithmetic, Duke Math J. 37 (1970), 439–445. MR274644

[St15A] D. Stanovský, A guide to self-distributive quasigroups, or Latin quandles, Quasigroups Related Systems 23 (2015), no. 1, 91–128. MR3353113

[St15T] —, The origins of involutory quandles, 2015. arXiv:1506.02389

Footnote